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Logarithmic Functions: Inverses, Rules, and Properties

Learn how to find the inverse of logarithmic functions and solve equations involving logarithmic and exponential forms. Explore the properties and rules of logarithmic functions.

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Logarithmic Functions: Inverses, Rules, and Properties

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  1. Warm-Up 10.2 Logarithmic Functions Find the inverse of each function. • f(x) = x + 10 • g(x) = 3x • h(x) = 5x + 3 • j(x) = ¼x + 2

  2. Logarithmic Functions 10.2 Logarithmic Functions • Write equivalent forms for exponential and logarithmic equations. • Use the definitions of exponential and logarithmic functions to solve equations.

  3. 10.2 Logarithmic Functions Rules and Properties Equivalent Exponential and Logarithmic Forms For any positive base b, where b 1: bx = y if and only if x= logby. Exponential form Logarithmic form

  4. Example 1 6.3 Logarithmic Functions a) Write 27 = 128 in logarithmic form. log2 128 = 7 b) Write log6 1296 = 4in exponential form. 64 = 1296

  5. Example 2 6.3 Logarithmic Functions a. Solve x = log2 8 for x. 2x = 8 x= 3 b. logx 25 = 2 x2 = 25 x= 5

  6. Practice 6.3 Logarithmic Functions c. Solve log2x = 4 for x. 24 = x x= 16

  7. Example 3 6.3 Logarithmic Functions a. Solve 10x = 14.5 for x. Round your answer to the nearest tenth. log1014.5 = x x= 1.161

  8. One-to-One Property of Exponential Functions Rules and Properties 6.3 Logarithmic Functions If bx= by, thenx = y.

  9. Example 4 6.3 Logarithmic Functions Find the value of the variable in each equation: b) log7 D= 3 a) log2 1 = r 73 = D 2r = 1 D = 343 20 = 1 r= 0

  10. Practice 6.3 Logarithmic Functions Find the value of the variable in each equation: 1) log4 64 = v 2) logv 25 = 2 3) 6 = log3v

  11. Practice Solve each equation for x. Round your answers to the nearest hundredth. 1) 10x = 1.498 2) 10x = 0.0054 Find the value of x in each equation. 3) x = log4 1 4) ½ = log9 x

  12. Properties of Logarithmic Functions Objectives: Simplify and evaluate expressions involving logarithms Solve equations involving logarithms

  13. Properties of Logarithms For m > 0, n > 0, b > 0, and b  1: Product Property logb (mn) = logb m + logb n

  14. Example 1 given: log5 12  1.5440 log5 10  1.4307 log5 120 = log5 (12)(10) = log5 12 + log5 10 1.5440 + 1.4307 2.9747

  15. logb = logb m – logb n m n Properties of Logarithms For m > 0, n > 0, b > 0, and b  1: Quotient Property

  16. 12 = log5 10 Example 2 given: log5 12  1.5440 log5 10  1.4307 log5 1.2 = log5 12 – log5 10 1.5440 – 1.4307 0.1133

  17. Properties of Logarithms For m > 0, n > 0, b > 0, and any real number p: Power Property logb mp = p logb m

  18. Example 3 given: log5 12  1.5440 log5 10  1.4307 log5 1254 5x = 125 = 4 log5 125 53 = 125 =4  3 x = 3 = 12

  19. Practice Write each expression as a single logarithm. 1) log2 14 – log2 7 2) log3 x + log3 4 – log3 2 3) 7 log3 y – 4 log3 x

  20. 4 minutes Warm-Up Write each expression as a single logarithm. Then simplify, if possible. 1) log6 6 + log6 30 – log6 5 2) log6 5x + 3(log6 x – log6 y)

  21. Properties of Logarithms For b > 0 and b  1: Exponential-Logarithmic Inverse Property logb bx = x and b logbx = x for x > 0

  22. Example 1 Evaluate each expression. a) b)

  23. Practice Evaluate each expression. 1) 7log711 – log3 81 2) log8 85 + 3log38

  24. Properties of Logarithms For b > 0 and b  1: One-to-One Property of Logarithms If logb x = logb y, then x = y

  25. Example 2 Solve log2(2x2 + 8x – 11) = log2(2x + 9) for x. log2(2x2 + 8x – 11) = log2(2x + 9) 2x2 + 8x – 11 = 2x + 9 2x2 + 6x – 20 = 0 2(x2 + 3x – 10) = 0 2(x – 2)(x + 5) = 0 x = -5,2 Check: log2(2x2 + 8x – 11) = log2(2x + 9) log2 (–1) = log2 (-1) undefined log2 13 = log2 13 true

  26. Practice Solve for x. 1) log5 (3x2 – 1) = log5 2x 2) logb (x2 – 2) + 2 logb 6 = logb 6x

  27. Exponential Growth and Decay Objectives: Determine the multiplier for exponential growth and decay Write and evaluate exponential expressions to model growth and decay situations

  28. Modeling Bacteria Growth Time (hr) 0 1 2 3 4 6 5 Population 25 50 100 200 400 1600 800 Write an algebraic expression that represents the population of bacteria after n hours. The expression is called an exponential expression because the exponent, n is a variable and the base, 2, is a fixed number. The base of an exponential expression is commonly referred to as the multiplier.

  29. Example 1 Find the multiplier for each rate of exponential growth or decay. a) 9% growth = 1.09 100% + 9% = 109% b) 0.08% growth = 1.0008 100% + 0.08% = 100.08% c) 2% decay = 0.98 100% - 2% = 98% d) 8.2% decay = 0.918 100% - 8.2% = 91.8%

  30. Example 2 Suppose that you invested $1000 in a company’s stock at the end of 1999 and that the value of the stock increased at a rate of about 15% per year. Predict the value of the stock, to the nearest cent, at the end of the years 2004 and 2009. Since the value of the stock is increasing at a rate of 15%, the multiplier will be 115%, or 1.15 = $2011.36 = $4045.56

  31. Example 3 Suppose that you buy a car for $15,000 and that its value decreases at a rate of about 8% per year. Predict the value of the car after 4 years and after 7 years. Since the value of the car is decreasing at a rate of 8%, the multiplier will be 92%, or 0.92 = $10,745.89 = $8,367.70

  32. Practice A vitamin is eliminated from the bloodstream at a rate of about 20% per hour. The vitamin reaches a peak level in the bloodstream of 300 mg. Predict the amount, to the nearest tenth of a milligram, of the vitamin remaining 2 hours after the peak level and 7 hours after the peak level.

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